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Let $\mathcal{C}$ be a nondegenerate planar curve and for a real, positive decreasing function $\psi$ let $\mathcal{C}(\psi)$ denote the set of simultaneously $\psi$-approximable points lying on $\mathcal{C}$. We show that $\mathcal{C}$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\mathcal{C}$ of $\mathcal{C}(\psi)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\mathcal{C}$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $\psi$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\mathcal{C}(\psi)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.
Victor Beresnevich 1 ; Detta Dickinson 2 ; Sanju Velani 3 ; R. C. Vaughan 4
@article{10_4007_annals_2007_166_367,
author = {Victor Beresnevich and Detta Dickinson and Sanju Velani and R. C. Vaughan},
title = {Diophantine approximation on planar curves and the distribution of rational points (with an {Appendix} by {R.} {C.} {Vaughan)}},
journal = {Annals of mathematics},
pages = {367--426},
publisher = {mathdoc},
volume = {166},
number = {2},
year = {2007},
doi = {10.4007/annals.2007.166.367},
mrnumber = {2373145},
zbl = {1137.11048},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.166.367/}
}
TY - JOUR AU - Victor Beresnevich AU - Detta Dickinson AU - Sanju Velani AU - R. C. Vaughan TI - Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan) JO - Annals of mathematics PY - 2007 SP - 367 EP - 426 VL - 166 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.166.367/ DO - 10.4007/annals.2007.166.367 LA - en ID - 10_4007_annals_2007_166_367 ER -
%0 Journal Article %A Victor Beresnevich %A Detta Dickinson %A Sanju Velani %A R. C. Vaughan %T Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan) %J Annals of mathematics %D 2007 %P 367-426 %V 166 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.166.367/ %R 10.4007/annals.2007.166.367 %G en %F 10_4007_annals_2007_166_367
Victor Beresnevich; Detta Dickinson; Sanju Velani; R. C. Vaughan. Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan). Annals of mathematics, Tome 166 (2007) no. 2, pp. 367-426. doi: 10.4007/annals.2007.166.367
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